On Binomial Sums and Alternating Binomial Sums of Generalized Fibonacci Numbers with Falling Factorials

Özet

In this paper, we consider and obtain binomial sums and alternating binomial sums including falling factorial of the summation indice. For example, For integers n and m such that 0 ≤ m < n, Xn k=0 n k k mU 2m 2k = n m (p 2 + 4)m Xm i=0 (−1)i 2m i V n−m 2(m−i) V2(m+n)(m−i) − (−1)m 2 n−m 2m m ! , and for positive odd integer m, Xn k=0 (−1)k n k k mV 2m k = n m mX−1 i=0 (−1)n(i+1) 2m i V n−m m−i V(m+n)(m−i) + 2m m 2 n−m !